Expectancy Score vs Sharpe Ratio

by Alex Matulich

Back when I set out to develop my own trading system, two
different successful traders recommended that I read Trade Your Way to Financial Freedom by Van K. Tharp. One trader
recommended it to me as a good book on position sizing. In spite
of its sensationalist title, it's a good book, for no other reason
than it contains a valuable feature: how to measure "quality" of
a trading strategy **objectively**, in terms of expectancy
multiplied by opportunity. I call this the "expectancy score."

Expectancy is how much you expect to earn from each trade for every dollar you risk. Opportunity is how often your strategy trades. You want to maximize the product of both.

*Expectancy*= (*AW*×*PW*+*AL*×*PL*) ⁄ |*AL*|- (expected profit per dollar risked)

*Expectancy score*=*Expectancy*×*Opportunity*

- where
*AW*= average winning trade (excluding maximum win)*PW*= probability of winning (*PW*=*<wins>*⁄*NST*

where*<wins>*is total wins excluding maximum win)*AL*= average losing trade (negative, excluding scratch losses)- |
*AL*| = absolute value of*AL* *PL*= probability of losing (*PL*=*<non-scratch losses>*⁄*NST*)*Opportunity*=*NST*× 365 ⁄*studydays*(opportunities to trade in a year)- where
*NST*=*<total trades>*−*<scratch trades>*− 1

In other words,*NST*= non-scratch trades during the period under test (a scratch trade loses commission+slippage or less) minus 1 (to exclude the maximum win).*studydays*= calendar days of history being tested

It is important to have the |*AL*| in the denominator of expectancy
because this converts the expectancy to "risk units" — earnings per
dollar risked.

This calculation of expectancy score, as described above, is different than that described by Tharp in three respects:

- First, I discard the maximum winning trade as an outlier.
For a system where the biggest win
*is*an outlier, discarding it will give a better representation of expectancy. For a system where the biggest win*isn't*an outlier, it won't matter other than to reduce the total winnings and total wins, leaving the average winnings largely unaffected. Discarding the largest win therefore gives a more conservative estimate of expectancy. - Second, I use the average loss rather than follow Tharp's recommendation to use the minimum loss as the unit of risk. The average loss is larger than the minimum loss and more likely to be experienced (average loss is typically near the peak of the distribution of losses), therefore using it will result in a more conservative estimate of expectancy than using the minimum loss. Tharp and I both exclude scratch losses (trades that lost only commission and slippage), because including these would bias the unit of risk too low.
- Third, I use the arithmetic average of wins and losses (after
deducting the largest win and scratch losses) as my
*expected*win and loss. Tharp has you create a histogram of winning trades and select a range that contains the peak of the curve. Tharp's way is not easily acheived through a computer algorithm because some subjectivity is required to determine the bin sizes of the histogram. My experiments show that the arithmetic average often falls at or near the histogram peak anyway, so that is what I use.

The expectancy score described above complements position sizing. You have to make a paradigm shift away from evaluating strategies based on net profit. Forget the net profit, forget drawdown, forget number of wins in a row, forget everything else Tradestation shows you in the Strategy Summary. These things mean nothing for strategy comparisons, because everyone has a subjective opinion about which of those measurements matter most.

In your mind you must decouple the entry/exit rules from "net profit" performance or "annualized return" performance. Instead, think of a strategy like this:

*Entry rules*control**risk**. Entries*don't*determine winners or losers!*Exit rules*determine**profits**or losses (winners or losers).*Entry and exit rules*together determine**expectancy**and**opportunity**.*Position sizing*determines your**net profit**or**return**, as well as**maximum drawdown**.

So when you're designing the entry/exit rules and their input
parameters, *don't* optimize for net profit! Instead . . .

Optimize for maximum expectancy score, without regard to anything else. Position sizing takes care of the rest. A good position sizing strategy will result in greater, more consistent profits on a high-expectancy strategy than on a low-expectancy strategy, even if the low-expectancy strategy has a higher net profit on a 1-contract basis!

Now, I know some trading software packages let you optimize strategy parameters based on anything you want. TradeStation gives you only canned results like net profit, win/loss ratio, drawdown, etc. For those of us who use TradeStation, I developed something that lets me optimize my strategies on expectancy score.

It's an EasyLanguage function (_SystemQuality). You just stick it at the end of your signal and start the optimizer. Every iteration of the optimizer will cause a line to be written to an Excel .csv file. Then all you do is load it into Excel, sort by the last column, and voila! The parameters for maximum expectancy score are right at the top.

The documentation included with the source code is detailed and should explain everything more fully. This function can be modified to use in optimizing anything else you want, also.

Some people like to use the Sharpe Ratio to gauge the relative quality of one trading strategy compared to another. After extensive research, I have no choice but to conclude that the Sharpe ratio isn't useful for objectively evaluating the merit of a system. It does have uses, but I do not agree that it should be used for determining overall merit.

Take two extremes for example:

- System
*A*returns 0.001% greater than the risk-free interest rate with zero drawdowns, and perfect consistency. - System
*B*returns 60% per year on your account with modest 10% drawdowns.

Which system would *you* rather trade? System *A* has
a higher Sharpe ratio — it's actually infinite due to zero standard
deviations in returns. Personally I'll take system *B* over
*A* any day! I am more concerned with my equity growth and
earning power of my risk capital, than whether periodic returns are
exactly the same.

All the Sharpe ratio does is measure consistency. True, that's one element of merit, but certainly not the whole picture. Using it to determine the merit of a whole trading strategy results in completely erroneous and subjective evaluations, as demonstrated by the extreme example above.

There's really only one objective way to measure the merit of a
system, and that's how much you expect it to earn for every dollar
risked combined with how often it gives you the opportunity to
earn that expected return. The risk concept is important; you're
measuring the return from your risk capital (i.e. your initial
stoploss), *not* what you actually "invest" in the market.

Develop a system that has a high expectancy score, and you'll find that the Sharpe ratio takes care of itself.

My research has led me down some fruitful paths, and some fruitless paths. Optimizing for Sharpe ratio is in the latter category.

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